A significant consequence of this formulation is that (REF ) clearly rules out the use of a genuine additive probability as the probabilistic predictor. (Our conjecture in Section REF  suggests a similar conclusion.) To see this most easily, take an extreme—albeit impractical—example where \(Y_{n+1}\)  is a real number and the assertion mapping \(A[y] = [y-\varepsilon , y+\varepsilon ]\) , with \(\varepsilon > 0\)  a small fixed value. Since a probability distribution \(\Pi _{Y^n}\)  on the real line would typically be absolutely continuous with respect to Lebesgue measure, it becomes virtually impossible for \(\Pi _{Y^n}(A[Y_{n+1}])\)  to be large, as a function of \((Y^n, Y_{n+1})\) . Therefore, the probability of the event “\(\Pi _{Y^n}(A[Y_{n+1}])\)  is small” would actually be large, hence (REF ) fails. Of course, it is not just the extreme examples of assertions of small size/measure one should be concerned about. Similar issues can arise due to the way probabilities are marginalized via integration, especially if the \(Y_i\) 's are multivariate. Stein's paradox [1]}, [2]} is a classical example of how correctly calibrated probabilities can become severely uncalibrated after marginalization, and what we are describing here is the prediction version of the false confidence phenomenon developed in [3]} [4]} in the context of statistical inference.
